LMFDB - Embedded newform 4009.2.a.c.1.51

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.51
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13354 q^{2} +1.40496 q^{3} -0.715079 q^{4} -3.22820 q^{5} +1.59258 q^{6} +1.93664 q^{7} -3.07766 q^{8} -1.02610 q^{9} +O(q^{10})$	\(q+1.13354 q^{2} +1.40496 q^{3} -0.715079 q^{4} -3.22820 q^{5} +1.59258 q^{6} +1.93664 q^{7} -3.07766 q^{8} -1.02610 q^{9} -3.65931 q^{10} +2.08980 q^{11} -1.00465 q^{12} -0.469607 q^{13} +2.19527 q^{14} -4.53549 q^{15} -2.05850 q^{16} +6.37626 q^{17} -1.16313 q^{18} +1.00000 q^{19} +2.30842 q^{20} +2.72090 q^{21} +2.36888 q^{22} +2.54145 q^{23} -4.32398 q^{24} +5.42131 q^{25} -0.532321 q^{26} -5.65649 q^{27} -1.38485 q^{28} -1.57718 q^{29} -5.14117 q^{30} -3.84807 q^{31} +3.82192 q^{32} +2.93608 q^{33} +7.22777 q^{34} -6.25188 q^{35} +0.733742 q^{36} -2.09082 q^{37} +1.13354 q^{38} -0.659778 q^{39} +9.93532 q^{40} -10.0172 q^{41} +3.08425 q^{42} -8.12692 q^{43} -1.49437 q^{44} +3.31246 q^{45} +2.88085 q^{46} -6.24105 q^{47} -2.89211 q^{48} -3.24942 q^{49} +6.14529 q^{50} +8.95837 q^{51} +0.335806 q^{52} -0.645267 q^{53} -6.41188 q^{54} -6.74631 q^{55} -5.96033 q^{56} +1.40496 q^{57} -1.78780 q^{58} -4.84961 q^{59} +3.24323 q^{60} -2.37183 q^{61} -4.36196 q^{62} -1.98719 q^{63} +8.44932 q^{64} +1.51599 q^{65} +3.32817 q^{66} +4.75672 q^{67} -4.55953 q^{68} +3.57063 q^{69} -7.08677 q^{70} -8.61439 q^{71} +3.15798 q^{72} +5.95088 q^{73} -2.37004 q^{74} +7.61669 q^{75} -0.715079 q^{76} +4.04720 q^{77} -0.747887 q^{78} -7.95295 q^{79} +6.64528 q^{80} -4.86882 q^{81} -11.3549 q^{82} -13.8061 q^{83} -1.94565 q^{84} -20.5839 q^{85} -9.21222 q^{86} -2.21586 q^{87} -6.43170 q^{88} +5.60457 q^{89} +3.75482 q^{90} -0.909461 q^{91} -1.81734 q^{92} -5.40637 q^{93} -7.07450 q^{94} -3.22820 q^{95} +5.36962 q^{96} +14.1822 q^{97} -3.68336 q^{98} -2.14434 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.13354	0.801536	0.400768	−	0.916179i	$-0.368743\pi$
$2$	1.13354	0.801536	0.400768	+	0.916179i	$0.368743\pi$
$3$	1.40496	0.811152	0.405576	−	0.914061i	$-0.367071\pi$
$3$	1.40496	0.811152	0.405576	+	0.914061i	$0.367071\pi$
$4$	−0.715079	−0.357539
$5$	−3.22820	−1.44370	−0.721849	−	0.692051i	$-0.756707\pi$
$5$	−3.22820	−1.44370	−0.721849	+	0.692051i	$0.756707\pi$
$6$	1.59258	0.650168
$7$	1.93664	0.731982	0.365991	−	0.930618i	$-0.380730\pi$
$7$	1.93664	0.731982	0.365991	+	0.930618i	$0.380730\pi$
$8$	−3.07766	−1.08812
$9$	−1.02610	−0.342033
$10$	−3.65931	−1.15718
$11$	2.08980	0.630099	0.315049	−	0.949075i	$-0.397979\pi$
$11$	2.08980	0.630099	0.315049	+	0.949075i	$0.397979\pi$
$12$	−1.00465	−0.290019
$13$	−0.469607	−0.130246	−0.0651228	−	0.997877i	$-0.520744\pi$
$13$	−0.469607	−0.130246	−0.0651228	+	0.997877i	$0.520744\pi$
$14$	2.19527	0.586710
$15$	−4.53549	−1.17106
$16$	−2.05850	−0.514626
$17$	6.37626	1.54647	0.773236	−	0.634119i	$-0.218637\pi$
$17$	6.37626	1.54647	0.773236	+	0.634119i	$0.218637\pi$
$18$	−1.16313	−0.274152
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.30842	0.516179
$21$	2.72090	0.593748
$22$	2.36888	0.505047
$23$	2.54145	0.529930	0.264965	−	0.964258i	$-0.414640\pi$
$23$	2.54145	0.529930	0.264965	+	0.964258i	$0.414640\pi$
$24$	−4.32398	−0.882628
$25$	5.42131	1.08426
$26$	−0.532321	−0.104397
$27$	−5.65649	−1.08859
$28$	−1.38485	−0.261712
$29$	−1.57718	−0.292875	−0.146437	−	0.989220i	$-0.546781\pi$
$29$	−1.57718	−0.292875	−0.146437	+	0.989220i	$0.546781\pi$
$30$	−5.14117	−0.938645
$31$	−3.84807	−0.691134	−0.345567	−	0.938394i	$-0.612313\pi$
$31$	−3.84807	−0.691134	−0.345567	+	0.938394i	$0.612313\pi$
$32$	3.82192	0.675626
$33$	2.93608	0.511106
$34$	7.22777	1.23955
$35$	−6.25188	−1.05676
$36$	0.733742	0.122290
$37$	−2.09082	−0.343729	−0.171865	−	0.985121i	$-0.554979\pi$
$37$	−2.09082	−0.343729	−0.171865	+	0.985121i	$0.554979\pi$
$38$	1.13354	0.183885
$39$	−0.659778	−0.105649
$40$	9.93532	1.57091
$41$	−10.0172	−1.56442	−0.782209	−	0.623016i	$-0.785907\pi$
$41$	−10.0172	−1.56442	−0.782209	+	0.623016i	$0.785907\pi$
$42$	3.08425	0.475911
$43$	−8.12692	−1.23934	−0.619672	−	0.784861i	$-0.712734\pi$
$43$	−8.12692	−1.23934	−0.619672	+	0.784861i	$0.712734\pi$
$44$	−1.49437	−0.225285
$45$	3.31246	0.493792
$46$	2.88085	0.424758
$47$	−6.24105	−0.910350	−0.455175	−	0.890402i	$-0.650423\pi$
$47$	−6.24105	−0.910350	−0.455175	+	0.890402i	$0.650423\pi$
$48$	−2.89211	−0.417440
$49$	−3.24942	−0.464203
$50$	6.14529	0.869075
$51$	8.95837	1.25442
$52$	0.335806	0.0465680
$53$	−0.645267	−0.0886342	−0.0443171	−	0.999018i	$-0.514111\pi$
$53$	−0.645267	−0.0886342	−0.0443171	+	0.999018i	$0.514111\pi$
$54$	−6.41188	−0.872546
$55$	−6.74631	−0.909672
$56$	−5.96033	−0.796482
$57$	1.40496	0.186091
$58$	−1.78780	−0.234750
$59$	−4.84961	−0.631365	−0.315683	−	0.948865i	$-0.602233\pi$
$59$	−4.84961	−0.631365	−0.315683	+	0.948865i	$0.602233\pi$
$60$	3.24323	0.418699
$61$	−2.37183	−0.303682	−0.151841	−	0.988405i	$-0.548520\pi$
$61$	−2.37183	−0.303682	−0.151841	+	0.988405i	$0.548520\pi$
$62$	−4.36196	−0.553969
$63$	−1.98719	−0.250362
$64$	8.44932	1.05616
$65$	1.51599	0.188035
$66$	3.32817	0.409670
$67$	4.75672	0.581126	0.290563	−	0.956856i	$-0.406157\pi$
$67$	4.75672	0.581126	0.290563	+	0.956856i	$0.406157\pi$
$68$	−4.55953	−0.552924
$69$	3.57063	0.429853
$70$	−7.08677	−0.847031
$71$	−8.61439	−1.02234	−0.511170	−	0.859480i	$-0.670788\pi$
$71$	−8.61439	−1.02234	−0.511170	+	0.859480i	$0.670788\pi$
$72$	3.15798	0.372172
$73$	5.95088	0.696498	0.348249	−	0.937402i	$-0.386776\pi$
$73$	5.95088	0.696498	0.348249	+	0.937402i	$0.386776\pi$
$74$	−2.37004	−0.275512
$75$	7.61669	0.879500
$76$	−0.715079	−0.0820252
$77$	4.04720	0.461221
$78$	−0.747887	−0.0846815
$79$	−7.95295	−0.894776	−0.447388	−	0.894340i	$-0.647646\pi$
$79$	−7.95295	−0.894776	−0.447388	+	0.894340i	$0.647646\pi$
$80$	6.64528	0.742964
$81$	−4.86882	−0.540980
$82$	−11.3549	−1.25394
$83$	−13.8061	−1.51542	−0.757709	−	0.652593i	$-0.773681\pi$
$83$	−13.8061	−1.51542	−0.757709	+	0.652593i	$0.773681\pi$
$84$	−1.94565	−0.212288
$85$	−20.5839	−2.23264
$86$	−9.21222	−0.993379
$87$	−2.21586	−0.237566
$88$	−6.43170	−0.685621
$89$	5.60457	0.594083	0.297042	−	0.954865i	$-0.404000\pi$
$89$	5.60457	0.594083	0.297042	+	0.954865i	$0.404000\pi$
$90$	3.75482	0.395792
$91$	−0.909461	−0.0953374
$92$	−1.81734	−0.189471
$93$	−5.40637	−0.560614
$94$	−7.07450	−0.729679
$95$	−3.22820	−0.331207
$96$	5.36962	0.548035
$97$	14.1822	1.43998	0.719991	−	0.693983i	$-0.244146\pi$
$97$	14.1822	1.43998	0.719991	+	0.693983i	$0.244146\pi$
$98$	−3.68336	−0.372075
$99$	−2.14434	−0.215515
$100$	−3.87666	−0.387666
$101$	−13.4540	−1.33872	−0.669362	−	0.742936i	$-0.733432\pi$
$101$	−13.4540	−1.33872	−0.669362	+	0.742936i	$0.733432\pi$
$102$	10.1547	1.00547
$103$	−17.0862	−1.68355	−0.841774	−	0.539830i	$-0.818489\pi$
$103$	−17.0862	−1.68355	−0.841774	+	0.539830i	$0.818489\pi$
$104$	1.44529	0.141723
$105$	−8.78361	−0.857192
$106$	−0.731438	−0.0710435
$107$	−7.98051	−0.771505	−0.385752	−	0.922602i	$-0.626058\pi$
$107$	−7.98051	−0.771505	−0.385752	+	0.922602i	$0.626058\pi$
$108$	4.04484	0.389215
$109$	8.29340	0.794364	0.397182	−	0.917740i	$-0.369988\pi$
$109$	8.29340	0.794364	0.397182	+	0.917740i	$0.369988\pi$
$110$	−7.64723	−0.729135
$111$	−2.93752	−0.278817
$112$	−3.98659	−0.376697
$113$	11.2668	1.05989	0.529946	−	0.848032i	$-0.322212\pi$
$113$	11.2668	1.05989	0.529946	+	0.848032i	$0.322212\pi$
$114$	1.59258	0.149159
$115$	−8.20433	−0.765058
$116$	1.12781	0.104714
$117$	0.481864	0.0445483
$118$	−5.49724	−0.506062
$119$	12.3485	1.13199
$120$	13.9587	1.27425
$121$	−6.63273	−0.602976
$122$	−2.68857	−0.243412
$123$	−14.0737	−1.26898
$124$	2.75167	0.247108
$125$	−1.36006	−0.121648
$126$	−2.25256	−0.200674
$127$	−19.2915	−1.71184	−0.855920	−	0.517109i	$-0.827008\pi$
$127$	−19.2915	−1.71184	−0.855920	+	0.517109i	$0.827008\pi$
$128$	1.93384	0.170929
$129$	−11.4180	−1.00530
$130$	1.71844	0.150717
$131$	−6.09585	−0.532597	−0.266298	−	0.963891i	$-0.585801\pi$
$131$	−6.09585	−0.532597	−0.266298	+	0.963891i	$0.585801\pi$
$132$	−2.09953	−0.182740
$133$	1.93664	0.167928
$134$	5.39195	0.465793
$135$	18.2603	1.57160
$136$	−19.6240	−1.68274
$137$	−20.4303	−1.74548	−0.872739	−	0.488187i	$-0.837658\pi$
$137$	−20.4303	−1.74548	−0.872739	+	0.488187i	$0.837658\pi$
$138$	4.04746	0.344543
$139$	20.2733	1.71955	0.859777	−	0.510669i	$-0.170602\pi$
$139$	20.2733	1.71955	0.859777	+	0.510669i	$0.170602\pi$
$140$	4.47058	0.377833
$141$	−8.76840	−0.738432
$142$	−9.76478	−0.819442
$143$	−0.981386	−0.0820676
$144$	2.11223	0.176019
$145$	5.09145	0.422822
$146$	6.74559	0.558269
$147$	−4.56529	−0.376539
$148$	1.49510	0.122897
$149$	12.1056	0.991730	0.495865	−	0.868400i	$-0.334851\pi$
$149$	12.1056	0.991730	0.495865	+	0.868400i	$0.334851\pi$
$150$	8.63386	0.704951
$151$	−5.81835	−0.473491	−0.236745	−	0.971572i	$-0.576081\pi$
$151$	−5.81835	−0.473491	−0.236745	+	0.971572i	$0.576081\pi$
$152$	−3.07766	−0.249631
$153$	−6.54268	−0.528944
$154$	4.58767	0.369685
$155$	12.4224	0.997788
$156$	0.471793	0.0377737
$157$	−8.02696	−0.640621	−0.320310	−	0.947313i	$-0.603787\pi$
$157$	−8.02696	−0.640621	−0.320310	+	0.947313i	$0.603787\pi$
$158$	−9.01501	−0.717196
$159$	−0.906571	−0.0718958
$160$	−12.3379	−0.975399
$161$	4.92188	0.387899
$162$	−5.51902	−0.433615
$163$	17.8500	1.39812	0.699060	−	0.715063i	$-0.253602\pi$
$163$	17.8500	1.39812	0.699060	+	0.715063i	$0.253602\pi$
$164$	7.16306	0.559341
$165$	−9.47826	−0.737882
$166$	−15.6498	−1.21466
$167$	19.4132	1.50224	0.751118	−	0.660168i	$-0.229515\pi$
$167$	19.4132	1.50224	0.751118	+	0.660168i	$0.229515\pi$
$168$	−8.37399	−0.646068
$169$	−12.7795	−0.983036
$170$	−23.3327	−1.78954
$171$	−1.02610	−0.0784678
$172$	5.81139	0.443114
$173$	18.8736	1.43494	0.717468	−	0.696592i	$-0.245301\pi$
$173$	18.8736	1.43494	0.717468	+	0.696592i	$0.245301\pi$
$174$	−2.51178	−0.190418
$175$	10.4991	0.793659
$176$	−4.30187	−0.324265
$177$	−6.81349	−0.512133
$178$	6.35302	0.476179
$179$	−16.2012	−1.21094	−0.605468	−	0.795870i	$-0.707014\pi$
$179$	−16.2012	−1.21094	−0.605468	+	0.795870i	$0.707014\pi$
$180$	−2.36867	−0.176550
$181$	−8.33692	−0.619678	−0.309839	−	0.950789i	$-0.600275\pi$
$181$	−8.33692	−0.619678	−0.309839	+	0.950789i	$0.600275\pi$
$182$	−1.03091	−0.0764164
$183$	−3.33232	−0.246332
$184$	−7.82173	−0.576626
$185$	6.74961	0.496241
$186$	−6.12835	−0.449353
$187$	13.3251	0.974430
$188$	4.46284	0.325486
$189$	−10.9546	−0.796830
$190$	−3.65931	−0.265474
$191$	16.3988	1.18658	0.593289	−	0.804990i	$-0.297829\pi$
$191$	16.3988	1.18658	0.593289	+	0.804990i	$0.297829\pi$
$192$	11.8709	0.856710
$193$	0.534428	0.0384690	0.0192345	−	0.999815i	$-0.493877\pi$
$193$	0.534428	0.0384690	0.0192345	+	0.999815i	$0.493877\pi$
$194$	16.0761	1.15420
$195$	2.12990	0.152525
$196$	2.32359	0.165971
$197$	−1.81608	−0.129391	−0.0646953	−	0.997905i	$-0.520608\pi$
$197$	−1.81608	−0.129391	−0.0646953	+	0.997905i	$0.520608\pi$
$198$	−2.43071	−0.172743
$199$	5.36812	0.380536	0.190268	−	0.981732i	$-0.439064\pi$
$199$	5.36812	0.380536	0.190268	+	0.981732i	$0.439064\pi$
$200$	−16.6849	−1.17980
$201$	6.68298	0.471381
$202$	−15.2507	−1.07304
$203$	−3.05443	−0.214379
$204$	−6.40594	−0.448506
$205$	32.3375	2.25855
$206$	−19.3679	−1.34943
$207$	−2.60778	−0.181253
$208$	0.966689	0.0670278
$209$	2.08980	0.144555
$210$	−9.95660	−0.687071
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	0.461416	0.0316902
$213$	−12.1028	−0.829272
$214$	−9.04625	−0.618389
$215$	26.2354	1.78924
$216$	17.4088	1.18452
$217$	−7.45233	−0.505897
$218$	9.40094	0.636712
$219$	8.36073	0.564966
$220$	4.82414	0.325243
$221$	−2.99434	−0.201421
$222$	−3.32980	−0.223482
$223$	6.33846	0.424454	0.212227	−	0.977220i	$-0.431928\pi$
$223$	6.33846	0.424454	0.212227	+	0.977220i	$0.431928\pi$
$224$	7.40168	0.494546
$225$	−5.56280	−0.370853
$226$	12.7714	0.849541
$227$	−20.5262	−1.36237	−0.681186	−	0.732111i	$-0.738535\pi$
$227$	−20.5262	−1.36237	−0.681186	+	0.732111i	$0.738535\pi$
$228$	−1.00465	−0.0665348
$229$	30.1489	1.99230	0.996148	−	0.0876851i	$-0.0279469\pi$
$229$	30.1489	1.99230	0.996148	+	0.0876851i	$0.0279469\pi$
$230$	−9.29997	−0.613222
$231$	5.68613	0.374120
$232$	4.85402	0.318682
$233$	9.86637	0.646367	0.323184	−	0.946336i	$-0.395247\pi$
$233$	9.86637	0.646367	0.323184	+	0.946336i	$0.395247\pi$
$234$	0.546214	0.0357071
$235$	20.1474	1.31427
$236$	3.46785	0.225738
$237$	−11.1735	−0.725799
$238$	13.9976	0.907330
$239$	−7.59522	−0.491294	−0.245647	−	0.969359i	$-0.579000\pi$
$239$	−7.59522	−0.491294	−0.245647	+	0.969359i	$0.579000\pi$
$240$	9.33632	0.602657
$241$	−3.74700	−0.241366	−0.120683	−	0.992691i	$-0.538508\pi$
$241$	−3.74700	−0.241366	−0.120683	+	0.992691i	$0.538508\pi$
$242$	−7.51849	−0.483307
$243$	10.1290	0.649775
$244$	1.69605	0.108578
$245$	10.4898	0.670168
$246$	−15.9531	−1.01713
$247$	−0.469607	−0.0298804
$248$	11.8431	0.752035
$249$	−19.3970	−1.22923
$250$	−1.54169	−0.0975049
$251$	25.9999	1.64110	0.820549	−	0.571576i	$-0.193668\pi$
$251$	25.9999	1.64110	0.820549	+	0.571576i	$0.193668\pi$
$252$	1.42099	0.0895142
$253$	5.31113	0.333908
$254$	−21.8677	−1.37210
$255$	−28.9195	−1.81101
$256$	−14.7065	−0.919159
$257$	11.5961	0.723346	0.361673	−	0.932305i	$-0.382206\pi$
$257$	11.5961	0.723346	0.361673	+	0.932305i	$0.382206\pi$
$258$	−12.9428	−0.805781
$259$	−4.04918	−0.251604
$260$	−1.08405	−0.0672300
$261$	1.61834	0.100173
$262$	−6.90991	−0.426896
$263$	−21.3505	−1.31653	−0.658264	−	0.752787i	$-0.728709\pi$
$263$	−21.3505	−1.31653	−0.658264	+	0.752787i	$0.728709\pi$
$264$	−9.03625	−0.556143
$265$	2.08305	0.127961
$266$	2.19527	0.134601
$267$	7.87417	0.481892
$268$	−3.40143	−0.207775
$269$	−2.03533	−0.124096	−0.0620480	−	0.998073i	$-0.519763\pi$
$269$	−2.03533	−0.124096	−0.0620480	+	0.998073i	$0.519763\pi$
$270$	20.6989	1.25969
$271$	−14.0716	−0.854786	−0.427393	−	0.904066i	$-0.640568\pi$
$271$	−14.0716	−0.854786	−0.427393	+	0.904066i	$0.640568\pi$
$272$	−13.1256	−0.795855
$273$	−1.27775	−0.0773331
$274$	−23.1586	−1.39906
$275$	11.3294	0.683192
$276$	−2.55328	−0.153689
$277$	−14.7801	−0.888047	−0.444024	−	0.896015i	$-0.646449\pi$
$277$	−14.7801	−0.888047	−0.444024	+	0.896015i	$0.646449\pi$
$278$	22.9806	1.37829
$279$	3.94850	0.236391
$280$	19.2411	1.14988
$281$	−9.40545	−0.561082	−0.280541	−	0.959842i	$-0.590514\pi$
$281$	−9.40545	−0.561082	−0.280541	+	0.959842i	$0.590514\pi$
$282$	−9.93936	−0.591880
$283$	−2.85859	−0.169925	−0.0849627	−	0.996384i	$-0.527077\pi$
$283$	−2.85859	−0.169925	−0.0849627	+	0.996384i	$0.527077\pi$
$284$	6.15996	0.365527
$285$	−4.53549	−0.268659
$286$	−1.11244	−0.0657802
$287$	−19.3997	−1.14513
$288$	−3.92166	−0.231086
$289$	23.6567	1.39157
$290$	5.77138	0.338907
$291$	19.9253	1.16804
$292$	−4.25535	−0.249026
$293$	−16.4524	−0.961157	−0.480579	−	0.876952i	$-0.659573\pi$
$293$	−16.4524	−0.961157	−0.480579	+	0.876952i	$0.659573\pi$
$294$	−5.17496	−0.301810
$295$	15.6555	0.911500
$296$	6.43485	0.374018
$297$	−11.8209	−0.685921
$298$	13.7222	0.794908
$299$	−1.19349	−0.0690210
$300$	−5.44654	−0.314456
$301$	−15.7389	−0.907177
$302$	−6.59536	−0.379520
$303$	−18.9023	−1.08591
$304$	−2.05850	−0.118063
$305$	7.65676	0.438425
$306$	−7.41641	−0.423968
$307$	−23.2536	−1.32715	−0.663577	−	0.748108i	$-0.730962\pi$
$307$	−23.2536	−1.32715	−0.663577	+	0.748108i	$0.730962\pi$
$308$	−2.89406	−0.164905
$309$	−24.0053	−1.36561
$310$	14.0813	0.799763
$311$	−11.6770	−0.662145	−0.331072	−	0.943605i	$-0.607410\pi$
$311$	−11.6770	−0.662145	−0.331072	+	0.943605i	$0.607410\pi$
$312$	2.03057	0.114958
$313$	−4.48098	−0.253280	−0.126640	−	0.991949i	$-0.540419\pi$
$313$	−4.48098	−0.253280	−0.126640	+	0.991949i	$0.540419\pi$
$314$	−9.09891	−0.513481
$315$	6.41504	0.361447
$316$	5.68698	0.319918
$317$	0.0996917	0.00559924	0.00279962	−	0.999996i	$-0.499109\pi$
$317$	0.0996917	0.00559924	0.00279962	+	0.999996i	$0.499109\pi$
$318$	−1.02764	−0.0576271
$319$	−3.29599	−0.184540
$320$	−27.2761	−1.52478
$321$	−11.2123	−0.625807
$322$	5.57917	0.310915
$323$	6.37626	0.354785
$324$	3.48159	0.193422
$325$	−2.54589	−0.141220
$326$	20.2338	1.12064
$327$	11.6519	0.644350
$328$	30.8294	1.70227
$329$	−12.0867	−0.666360
$330$	−10.7440	−0.591439
$331$	21.7798	1.19713	0.598563	−	0.801076i	$-0.295739\pi$
$331$	21.7798	1.19713	0.598563	+	0.801076i	$0.295739\pi$
$332$	9.87246	0.541822
$333$	2.14539	0.117567
$334$	22.0057	1.20410
$335$	−15.3557	−0.838969
$336$	−5.60098	−0.305558
$337$	10.9780	0.598008	0.299004	−	0.954252i	$-0.403346\pi$
$337$	10.9780	0.598008	0.299004	+	0.954252i	$0.403346\pi$
$338$	−14.4861	−0.787939
$339$	15.8294	0.859732
$340$	14.7191	0.798255
$341$	−8.04170	−0.435482
$342$	−1.16313	−0.0628948
$343$	−19.8495	−1.07177
$344$	25.0119	1.34855
$345$	−11.5267	−0.620578
$346$	21.3941	1.15015
$347$	27.1171	1.45572	0.727860	−	0.685726i	$-0.240515\pi$
$347$	27.1171	1.45572	0.727860	+	0.685726i	$0.240515\pi$
$348$	1.58452	0.0849391
$349$	21.0376	1.12612	0.563059	−	0.826417i	$-0.309624\pi$
$349$	21.0376	1.12612	0.563059	+	0.826417i	$0.309624\pi$
$350$	11.9012	0.636147
$351$	2.65633	0.141784
$352$	7.98704	0.425711
$353$	3.75762	0.199998	0.0999990	−	0.994988i	$-0.468116\pi$
$353$	3.75762	0.199998	0.0999990	+	0.994988i	$0.468116\pi$
$354$	−7.72338	−0.410493
$355$	27.8090	1.47595
$356$	−4.00771	−0.212408
$357$	17.3492	0.918214
$358$	−18.3648	−0.970610
$359$	−6.35203	−0.335248	−0.167624	−	0.985851i	$-0.553609\pi$
$359$	−6.35203	−0.335248	−0.167624	+	0.985851i	$0.553609\pi$
$360$	−10.1946	−0.537304
$361$	1.00000	0.0526316
$362$	−9.45026	−0.496695
$363$	−9.31870	−0.489105
$364$	0.650336	0.0340869
$365$	−19.2107	−1.00553
$366$	−3.77733	−0.197444
$367$	−12.4766	−0.651275	−0.325637	−	0.945495i	$-0.605579\pi$
$367$	−12.4766	−0.651275	−0.325637	+	0.945495i	$0.605579\pi$
$368$	−5.23159	−0.272716
$369$	10.2786	0.535083
$370$	7.65098	0.397755
$371$	−1.24965	−0.0648786
$372$	3.86598	0.200442
$373$	2.51244	0.130089	0.0650447	−	0.997882i	$-0.479281\pi$
$373$	2.51244	0.130089	0.0650447	+	0.997882i	$0.479281\pi$
$374$	15.1046	0.781041
$375$	−1.91083	−0.0986746
$376$	19.2078	0.990568
$377$	0.740654	0.0381456
$378$	−12.4175	−0.638688
$379$	−17.3540	−0.891413	−0.445707	−	0.895179i	$-0.647048\pi$
$379$	−17.3540	−0.891413	−0.445707	+	0.895179i	$0.647048\pi$
$380$	2.30842	0.118419
$381$	−27.1036	−1.38856
$382$	18.5888	0.951085
$383$	−31.3733	−1.60310	−0.801550	−	0.597928i	$-0.795991\pi$
$383$	−31.3733	−1.60310	−0.801550	+	0.597928i	$0.795991\pi$
$384$	2.71696	0.138649
$385$	−13.0652	−0.665863
$386$	0.605797	0.0308343
$387$	8.33903	0.423896
$388$	−10.1414	−0.514850
$389$	−2.98903	−0.151550	−0.0757748	−	0.997125i	$-0.524143\pi$
$389$	−2.98903	−0.151550	−0.0757748	+	0.997125i	$0.524143\pi$
$390$	2.41433	0.122254
$391$	16.2050	0.819521
$392$	10.0006	0.505107
$393$	−8.56440	−0.432017
$394$	−2.05861	−0.103711
$395$	25.6737	1.29179
$396$	1.53337	0.0770549
$397$	−2.29697	−0.115281	−0.0576407	−	0.998337i	$-0.518358\pi$
$397$	−2.29697	−0.115281	−0.0576407	+	0.998337i	$0.518358\pi$
$398$	6.08500	0.305014
$399$	2.72090	0.136215
$400$	−11.1598	−0.557989
$401$	−15.3435	−0.766218	−0.383109	−	0.923703i	$-0.625147\pi$
$401$	−15.3435	−0.766218	−0.383109	+	0.923703i	$0.625147\pi$
$402$	7.57545	0.377829
$403$	1.80708	0.0900172
$404$	9.62068	0.478647
$405$	15.7176	0.781012
$406$	−3.46233	−0.171832
$407$	−4.36941	−0.216583
$408$	−27.5708	−1.36496
$409$	13.4596	0.665536	0.332768	−	0.943009i	$-0.392017\pi$
$409$	13.4596	0.665536	0.332768	+	0.943009i	$0.392017\pi$
$410$	36.6559	1.81031
$411$	−28.7037	−1.41585
$412$	12.2179	0.601935
$413$	−9.39195	−0.462148
$414$	−2.95604	−0.145281
$415$	44.5690	2.18780
$416$	−1.79480	−0.0879973
$417$	28.4830	1.39482
$418$	2.36888	0.115866
$419$	10.5115	0.513522	0.256761	−	0.966475i	$-0.417345\pi$
$419$	10.5115	0.513522	0.256761	+	0.966475i	$0.417345\pi$
$420$	6.28097	0.306480
$421$	−5.90107	−0.287600	−0.143800	−	0.989607i	$-0.545932\pi$
$421$	−5.90107	−0.287600	−0.143800	+	0.989607i	$0.545932\pi$
$422$	1.13354	0.0551800
$423$	6.40393	0.311370
$424$	1.98591	0.0964444
$425$	34.5677	1.67678
$426$	−13.7191	−0.664692
$427$	−4.59339	−0.222290
$428$	5.70669	0.275843
$429$	−1.37880	−0.0665693
$430$	29.7389	1.43414
$431$	−23.5251	−1.13316	−0.566582	−	0.824005i	$-0.691735\pi$
$431$	−23.5251	−1.13316	−0.566582	+	0.824005i	$0.691735\pi$
$432$	11.6439	0.560218
$433$	15.5882	0.749121	0.374560	−	0.927203i	$-0.377794\pi$
$433$	15.5882	0.749121	0.374560	+	0.927203i	$0.377794\pi$
$434$	−8.44754	−0.405495
$435$	7.15327	0.342973
$436$	−5.93044	−0.284016
$437$	2.54145	0.121574
$438$	9.47725	0.452841
$439$	−32.8174	−1.56629	−0.783146	−	0.621838i	$-0.786386\pi$
$439$	−32.8174	−1.56629	−0.783146	+	0.621838i	$0.786386\pi$
$440$	20.7628	0.989829
$441$	3.33423	0.158773
$442$	−3.39422	−0.161446
$443$	8.54957	0.406202	0.203101	−	0.979158i	$-0.434898\pi$
$443$	8.54957	0.406202	0.203101	+	0.979158i	$0.434898\pi$
$444$	2.10056	0.0996879
$445$	−18.0927	−0.857676
$446$	7.18492	0.340216
$447$	17.0078	0.804444
$448$	16.3633	0.773093
$449$	8.20517	0.387226	0.193613	−	0.981078i	$-0.437979\pi$
$449$	8.20517	0.387226	0.193613	+	0.981078i	$0.437979\pi$
$450$	−6.30567	−0.297252
$451$	−20.9339	−0.985738
$452$	−8.05665	−0.378953
$453$	−8.17453	−0.384073
$454$	−23.2673	−1.09199
$455$	2.93593	0.137638
$456$	−4.32398	−0.202489
$457$	0.192036	0.00898305	0.00449152	−	0.999990i	$-0.498570\pi$
$457$	0.192036	0.00898305	0.00449152	+	0.999990i	$0.498570\pi$
$458$	34.1751	1.59690
$459$	−36.0673	−1.68348
$460$	5.86674	0.273538
$461$	−39.4141	−1.83570	−0.917849	−	0.396930i	$-0.870075\pi$
$461$	−39.4141	−1.83570	−0.917849	+	0.396930i	$0.870075\pi$
$462$	6.44548	0.299871
$463$	28.8414	1.34038	0.670188	−	0.742191i	$-0.266214\pi$
$463$	28.8414	1.34038	0.670188	+	0.742191i	$0.266214\pi$
$464$	3.24663	0.150721
$465$	17.4529	0.809357
$466$	11.1840	0.518087
$467$	−24.4860	−1.13308	−0.566539	−	0.824035i	$-0.691718\pi$
$467$	−24.4860	−1.13308	−0.566539	+	0.824035i	$0.691718\pi$
$468$	−0.344571	−0.0159278
$469$	9.21206	0.425373
$470$	22.8379	1.05344
$471$	−11.2775	−0.519641
$472$	14.9254	0.686999
$473$	−16.9836	−0.780909
$474$	−12.6657	−0.581755
$475$	5.42131	0.248747
$476$	−8.83018	−0.404731
$477$	0.662107	0.0303158
$478$	−8.60951	−0.393790
$479$	34.1085	1.55845	0.779227	−	0.626741i	$-0.215612\pi$
$479$	34.1085	1.55845	0.779227	+	0.626741i	$0.215612\pi$
$480$	−17.3342	−0.791196
$481$	0.981867	0.0447693
$482$	−4.24739	−0.193463
$483$	6.91503	0.314645
$484$	4.74293	0.215588
$485$	−45.7830	−2.07890
$486$	11.4817	0.520818
$487$	26.5033	1.20098	0.600490	−	0.799632i	$-0.294972\pi$
$487$	26.5033	1.20098	0.600490	+	0.799632i	$0.294972\pi$
$488$	7.29969	0.330441
$489$	25.0785	1.13409
$490$	11.8906	0.537164
$491$	−19.4904	−0.879590	−0.439795	−	0.898098i	$-0.644949\pi$
$491$	−19.4904	−0.879590	−0.439795	+	0.898098i	$0.644949\pi$
$492$	10.0638	0.453711
$493$	−10.0565	−0.452922
$494$	−0.532321	−0.0239502
$495$	6.92238	0.311138
$496$	7.92127	0.355676
$497$	−16.6830	−0.748334
$498$	−21.9873	−0.985276
$499$	35.5008	1.58923	0.794617	−	0.607111i	$-0.207672\pi$
$499$	35.5008	1.58923	0.794617	+	0.607111i	$0.207672\pi$
$500$	0.972550	0.0434938
$501$	27.2747	1.21854
$502$	29.4720	1.31540
$503$	9.74876	0.434676	0.217338	−	0.976096i	$-0.430263\pi$
$503$	9.74876	0.434676	0.217338	+	0.976096i	$0.430263\pi$
$504$	6.11588	0.272423
$505$	43.4323	1.93271
$506$	6.02040	0.267639
$507$	−17.9546	−0.797391
$508$	13.7949	0.612050
$509$	−34.7177	−1.53884	−0.769418	−	0.638746i	$-0.779453\pi$
$509$	−34.7177	−1.53884	−0.769418	+	0.638746i	$0.779453\pi$
$510$	−32.7815	−1.45159
$511$	11.5247	0.509824
$512$	−20.5382	−0.907668
$513$	−5.65649	−0.249740
$514$	13.1447	0.579788
$515$	55.1576	2.43053
$516$	8.16474	0.359433
$517$	−13.0425	−0.573611
$518$	−4.58992	−0.201669
$519$	26.5166	1.16395
$520$	−4.66570	−0.204604
$521$	2.58138	0.113092	0.0565461	−	0.998400i	$-0.481991\pi$
$521$	2.58138	0.113092	0.0565461	+	0.998400i	$0.481991\pi$
$522$	1.83446	0.0802921
$523$	30.8277	1.34800	0.674001	−	0.738731i	$-0.264574\pi$
$523$	30.8277	1.34800	0.674001	+	0.738731i	$0.264574\pi$
$524$	4.35901	0.190424
$525$	14.7508	0.643778
$526$	−24.2017	−1.05525
$527$	−24.5363	−1.06882
$528$	−6.04393	−0.263028
$529$	−16.5410	−0.719175
$530$	2.36123	0.102565
$531$	4.97618	0.215948
$532$	−1.38485	−0.0600409
$533$	4.70414	0.203759
$534$	8.92572	0.386254
$535$	25.7627	1.11382
$536$	−14.6396	−0.632333
$537$	−22.7620	−0.982253
$538$	−2.30713	−0.0994675
$539$	−6.79064	−0.292494
$540$	−13.0576	−0.561908
$541$	32.3922	1.39265	0.696325	−	0.717726i	$-0.254817\pi$
$541$	32.3922	1.39265	0.696325	+	0.717726i	$0.254817\pi$
$542$	−15.9507	−0.685142
$543$	−11.7130	−0.502653
$544$	24.3695	1.04484
$545$	−26.7728	−1.14682
$546$	−1.44839	−0.0619853
$547$	6.59067	0.281797	0.140898	−	0.990024i	$-0.455001\pi$
$547$	6.59067	0.281797	0.140898	+	0.990024i	$0.455001\pi$
$548$	14.6093	0.624077
$549$	2.43373	0.103869
$550$	12.8424	0.547603
$551$	−1.57718	−0.0671900
$552$	−10.9892	−0.467731
$553$	−15.4020	−0.654960
$554$	−16.7538	−0.711802
$555$	9.48290	0.402527
$556$	−14.4970	−0.614809
$557$	−33.4822	−1.41869	−0.709344	−	0.704863i	$-0.751008\pi$
$557$	−33.4822	−1.41869	−0.709344	+	0.704863i	$0.751008\pi$
$558$	4.47580	0.189476
$559$	3.81646	0.161419
$560$	12.8695	0.543836
$561$	18.7212	0.790410
$562$	−10.6615	−0.449728
$563$	−22.6121	−0.952984	−0.476492	−	0.879179i	$-0.658092\pi$
$563$	−22.6121	−0.952984	−0.476492	+	0.879179i	$0.658092\pi$
$564$	6.27009	0.264019
$565$	−36.3715	−1.53016
$566$	−3.24033	−0.136201
$567$	−9.42917	−0.395988
$568$	26.5122	1.11243
$569$	−36.2888	−1.52131	−0.760653	−	0.649159i	$-0.775121\pi$
$569$	−36.2888	−1.52131	−0.760653	+	0.649159i	$0.775121\pi$
$570$	−5.14117	−0.215340
$571$	36.0912	1.51037	0.755185	−	0.655512i	$-0.227547\pi$
$571$	36.0912	1.51037	0.755185	+	0.655512i	$0.227547\pi$
$572$	0.701768	0.0293424
$573$	23.0396	0.962495
$574$	−21.9904	−0.917860
$575$	13.7780	0.574582
$576$	−8.66984	−0.361243
$577$	−24.3288	−1.01282	−0.506412	−	0.862292i	$-0.669028\pi$
$577$	−24.3288	−1.01282	−0.506412	+	0.862292i	$0.669028\pi$
$578$	26.8160	1.11540
$579$	0.750848	0.0312042
$580$	−3.64079	−0.151176
$581$	−26.7375	−1.10926
$582$	22.5862	0.936230
$583$	−1.34848	−0.0558483
$584$	−18.3148	−0.757872
$585$	−1.55555	−0.0643143
$586$	−18.6495	−0.770402
$587$	12.7708	0.527105	0.263553	−	0.964645i	$-0.415106\pi$
$587$	12.7708	0.527105	0.263553	+	0.964645i	$0.415106\pi$
$588$	3.26454	0.134627
$589$	−3.84807	−0.158557
$590$	17.7462	0.730601
$591$	−2.55152	−0.104955
$592$	4.30397	0.176892
$593$	26.1877	1.07540	0.537700	−	0.843136i	$-0.319293\pi$
$593$	26.1877	1.07540	0.537700	+	0.843136i	$0.319293\pi$
$594$	−13.3996	−0.549790
$595$	−39.8636	−1.63425
$596$	−8.65646	−0.354583
$597$	7.54198	0.308673
$598$	−1.35287	−0.0553229
$599$	41.9526	1.71414	0.857068	−	0.515204i	$-0.172284\pi$
$599$	41.9526	1.71414	0.857068	+	0.515204i	$0.172284\pi$
$600$	−23.4416	−0.956999
$601$	−46.7237	−1.90590	−0.952949	−	0.303131i	$-0.901968\pi$
$601$	−46.7237	−1.90590	−0.952949	+	0.303131i	$0.901968\pi$
$602$	−17.8408	−0.727135
$603$	−4.88086	−0.198764
$604$	4.16058	0.169292
$605$	21.4118	0.870514
$606$	−21.4266	−0.870395
$607$	13.6217	0.552889	0.276444	−	0.961030i	$-0.410844\pi$
$607$	13.6217	0.552889	0.276444	+	0.961030i	$0.410844\pi$
$608$	3.82192	0.154999
$609$	−4.29134	−0.173894
$610$	8.67927	0.351413
$611$	2.93084	0.118569
$612$	4.67853	0.189118
$613$	−24.1163	−0.974047	−0.487024	−	0.873389i	$-0.661917\pi$
$613$	−24.1163	−0.974047	−0.487024	+	0.873389i	$0.661917\pi$
$614$	−26.3590	−1.06376
$615$	45.4327	1.83202
$616$	−12.4559	−0.501862
$617$	15.6668	0.630721	0.315360	−	0.948972i	$-0.397875\pi$
$617$	15.6668	0.630721	0.315360	+	0.948972i	$0.397875\pi$
$618$	−27.2110	−1.09459
$619$	8.59362	0.345407	0.172703	−	0.984974i	$-0.444750\pi$
$619$	8.59362	0.345407	0.172703	+	0.984974i	$0.444750\pi$
$620$	−8.88296	−0.356748
$621$	−14.3757	−0.576877
$622$	−13.2364	−0.530733
$623$	10.8540	0.434858
$624$	1.35816	0.0543697
$625$	−22.7160	−0.908639
$626$	−5.07938	−0.203013
$627$	2.93608	0.117256
$628$	5.73991	0.229047
$629$	−13.3316	−0.531568
$630$	7.27173	0.289713
$631$	23.0495	0.917587	0.458794	−	0.888543i	$-0.348282\pi$
$631$	23.0495	0.917587	0.458794	+	0.888543i	$0.348282\pi$
$632$	24.4765	0.973621
$633$	1.40496	0.0558420
$634$	0.113005	0.00448800
$635$	62.2768	2.47138
$636$	0.648270	0.0257056
$637$	1.52595	0.0604604
$638$	−3.73615	−0.147915
$639$	8.83921	0.349674
$640$	−6.24283	−0.246770
$641$	16.4222	0.648639	0.324320	−	0.945948i	$-0.394865\pi$
$641$	16.4222	0.648639	0.324320	+	0.945948i	$0.394865\pi$
$642$	−12.7096	−0.501607
$643$	−25.3254	−0.998736	−0.499368	−	0.866390i	$-0.666434\pi$
$643$	−25.3254	−0.998736	−0.499368	+	0.866390i	$0.666434\pi$
$644$	−3.51953	−0.138689
$645$	36.8595	1.45134
$646$	7.22777	0.284373
$647$	−26.3564	−1.03618	−0.518089	−	0.855327i	$-0.673356\pi$
$647$	−26.3564	−1.03618	−0.518089	+	0.855327i	$0.673356\pi$
$648$	14.9846	0.588650
$649$	−10.1347	−0.397822
$650$	−2.88587	−0.113193
$651$	−10.4702	−0.410359
$652$	−12.7642	−0.499883
$653$	−4.25856	−0.166650	−0.0833252	−	0.996522i	$-0.526554\pi$
$653$	−4.25856	−0.166650	−0.0833252	+	0.996522i	$0.526554\pi$
$654$	13.2079	0.516470
$655$	19.6786	0.768908
$656$	20.6204	0.805091
$657$	−6.10620	−0.238225
$658$	−13.7008	−0.534112
$659$	−8.79547	−0.342623	−0.171311	−	0.985217i	$-0.554800\pi$
$659$	−8.79547	−0.342623	−0.171311	+	0.985217i	$0.554800\pi$
$660$	6.77770	0.263822
$661$	15.0197	0.584200	0.292100	−	0.956388i	$-0.405646\pi$
$661$	15.0197	0.584200	0.292100	+	0.956388i	$0.405646\pi$
$662$	24.6884	0.959540
$663$	−4.20692	−0.163383
$664$	42.4905	1.64895
$665$	−6.25188	−0.242437
$666$	2.43190	0.0942341
$667$	−4.00832	−0.155203
$668$	−13.8820	−0.537109
$669$	8.90525	0.344297
$670$	−17.4063	−0.672464
$671$	−4.95665	−0.191350
$672$	10.3990	0.401151
$673$	−16.1510	−0.622577	−0.311288	−	0.950315i	$-0.600761\pi$
$673$	−16.1510	−0.622577	−0.311288	+	0.950315i	$0.600761\pi$
$674$	12.4440	0.479325
$675$	−30.6656	−1.18032
$676$	9.13833	0.351474
$677$	17.3088	0.665232	0.332616	−	0.943062i	$-0.392069\pi$
$677$	17.3088	0.665232	0.332616	+	0.943062i	$0.392069\pi$
$678$	17.9433	0.689107
$679$	27.4658	1.05404
$680$	63.3502	2.42937
$681$	−28.8384	−1.10509
$682$	−9.11562	−0.349055
$683$	16.9005	0.646680	0.323340	−	0.946283i	$-0.395194\pi$
$683$	16.9005	0.646680	0.323340	+	0.946283i	$0.395194\pi$
$684$	0.733742	0.0280553
$685$	65.9532	2.51994
$686$	−22.5002	−0.859062
$687$	42.3579	1.61605
$688$	16.7293	0.637799
$689$	0.303022	0.0115442
$690$	−13.0660	−0.497416
$691$	33.3013	1.26684	0.633421	−	0.773808i	$-0.281650\pi$
$691$	33.3013	1.26684	0.633421	+	0.773808i	$0.281650\pi$
$692$	−13.4961	−0.513046
$693$	−4.15282	−0.157753
$694$	30.7384	1.16681
$695$	−65.4462	−2.48252
$696$	6.81968	0.258499
$697$	−63.8721	−2.41933
$698$	23.8470	0.902624
$699$	13.8618	0.524302
$700$	−7.50770	−0.283764
$701$	9.97099	0.376599	0.188300	−	0.982112i	$-0.439702\pi$
$701$	9.97099	0.376599	0.188300	+	0.982112i	$0.439702\pi$
$702$	3.01107	0.113645
$703$	−2.09082	−0.0788569
$704$	17.6574	0.665488
$705$	28.3062	1.06607
$706$	4.25943	0.160306
$707$	−26.0556	−0.979922
$708$	4.87218	0.183108
$709$	48.9623	1.83882	0.919408	−	0.393304i	$-0.128668\pi$
$709$	48.9623	1.83882	0.919408	+	0.393304i	$0.128668\pi$
$710$	31.5227	1.18303
$711$	8.16051	0.306043
$712$	−17.2490	−0.646432
$713$	−9.77969	−0.366252
$714$	19.6660	0.735982
$715$	3.16811	0.118481
$716$	11.5852	0.432957
$717$	−10.6709	−0.398514
$718$	−7.20031	−0.268713
$719$	−13.8649	−0.517073	−0.258537	−	0.966001i	$-0.583240\pi$
$719$	−13.8649	−0.517073	−0.258537	+	0.966001i	$0.583240\pi$
$720$	−6.81871	−0.254118
$721$	−33.0898	−1.23233
$722$	1.13354	0.0421861
$723$	−5.26437	−0.195784
$724$	5.96155	0.221559
$725$	−8.55036	−0.317552
$726$	−10.5631	−0.392035
$727$	−25.8981	−0.960506	−0.480253	−	0.877130i	$-0.659455\pi$
$727$	−25.8981	−0.960506	−0.480253	+	0.877130i	$0.659455\pi$
$728$	2.79901	0.103738
$729$	28.8373	1.06805
$730$	−21.7761	−0.805971
$731$	−51.8194	−1.91661
$732$	2.38287	0.0880734
$733$	1.34832	0.0498013	0.0249007	−	0.999690i	$-0.492073\pi$
$733$	1.34832	0.0498013	0.0249007	+	0.999690i	$0.492073\pi$
$734$	−14.1428	−0.522020
$735$	14.7377	0.543608
$736$	9.71322	0.358034
$737$	9.94059	0.366166
$738$	11.6512	0.428888
$739$	52.2342	1.92147	0.960733	−	0.277474i	$-0.0894971\pi$
$739$	52.2342	1.92147	0.960733	+	0.277474i	$0.0894971\pi$
$740$	−4.82650	−0.177426
$741$	−0.659778	−0.0242375
$742$	−1.41653	−0.0520026
$743$	−41.3189	−1.51584	−0.757921	−	0.652346i	$-0.773785\pi$
$743$	−41.3189	−1.51584	−0.757921	+	0.652346i	$0.773785\pi$
$744$	16.6390	0.610014
$745$	−39.0794	−1.43176
$746$	2.84796	0.104271
$747$	14.1664	0.518323
$748$	−9.52851	−0.348397
$749$	−15.4554	−0.564727
$750$	−2.16600	−0.0790913
$751$	10.1885	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$751$	10.1885	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$752$	12.8472	0.468490
$753$	36.5287	1.33118
$754$	0.839564	0.0305751
$755$	18.7828	0.683577
$756$	7.83340	0.284898
$757$	26.7704	0.972986	0.486493	−	0.873684i	$-0.338276\pi$
$757$	26.7704	0.972986	0.486493	+	0.873684i	$0.338276\pi$
$758$	−19.6715	−0.714500
$759$	7.46190	0.270850
$760$	9.93532	0.360392
$761$	−24.9341	−0.903859	−0.451930	−	0.892054i	$-0.649264\pi$
$761$	−24.9341	−0.903859	−0.451930	+	0.892054i	$0.649264\pi$
$762$	−30.7232	−1.11298
$763$	16.0614	0.581460
$764$	−11.7265	−0.424248
$765$	21.1211	0.763635
$766$	−35.5630	−1.28494
$767$	2.27741	0.0822326
$768$	−20.6620	−0.745577
$769$	−2.93397	−0.105802	−0.0529009	−	0.998600i	$-0.516847\pi$
$769$	−2.93397	−0.105802	−0.0529009	+	0.998600i	$0.516847\pi$
$770$	−14.8099	−0.533713
$771$	16.2920	0.586743
$772$	−0.382158	−0.0137542
$773$	−23.3453	−0.839671	−0.419836	−	0.907600i	$-0.637912\pi$
$773$	−23.3453	−0.839671	−0.419836	+	0.907600i	$0.637912\pi$
$774$	9.45265	0.339768
$775$	−20.8616	−0.749369
$776$	−43.6479	−1.56687
$777$	−5.68892	−0.204089
$778$	−3.38819	−0.121473
$779$	−10.0172	−0.358902
$780$	−1.52304	−0.0545337
$781$	−18.0024	−0.644175
$782$	18.3690	0.656876
$783$	8.92129	0.318821
$784$	6.68895	0.238891
$785$	25.9127	0.924862
$786$	−9.70812	−0.346277
$787$	−36.6019	−1.30472	−0.652359	−	0.757910i	$-0.726220\pi$
$787$	−36.6019	−1.30472	−0.652359	+	0.757910i	$0.726220\pi$
$788$	1.29864	0.0462622
$789$	−29.9965	−1.06790
$790$	29.1023	1.03541
$791$	21.8197	0.775821
$792$	6.59956	0.234505
$793$	1.11383	0.0395532
$794$	−2.60371	−0.0924022
$795$	2.92660	0.103796
$796$	−3.83863	−0.136057
$797$	24.9805	0.884853	0.442427	−	0.896805i	$-0.354118\pi$
$797$	24.9805	0.884853	0.442427	+	0.896805i	$0.354118\pi$
$798$	3.08425	0.109181
$799$	−39.7946	−1.40783
$800$	20.7198	0.732555
$801$	−5.75084	−0.203196
$802$	−17.3925	−0.614151
$803$	12.4362	0.438863
$804$	−4.77886	−0.168537
$805$	−15.8888	−0.560008
$806$	2.04841	0.0721520
$807$	−2.85954	−0.100661
$808$	41.4069	1.45669
$809$	−42.2052	−1.48385	−0.741927	−	0.670480i	$-0.766088\pi$
$809$	−42.2052	−1.48385	−0.741927	+	0.670480i	$0.766088\pi$
$810$	17.8165	0.626009
$811$	27.6374	0.970479	0.485239	−	0.874381i	$-0.338732\pi$
$811$	27.6374	0.970479	0.485239	+	0.874381i	$0.338732\pi$
$812$	2.18416	0.0766489
$813$	−19.7699	−0.693361
$814$	−4.95291	−0.173600
$815$	−57.6235	−2.01846
$816$	−18.4408	−0.645559
$817$	−8.12692	−0.284325
$818$	15.2571	0.533451
$819$	0.933197	0.0326086
$820$	−23.1238	−0.807519
$821$	−44.0603	−1.53771	−0.768857	−	0.639421i	$-0.779174\pi$
$821$	−44.0603	−1.53771	−0.768857	+	0.639421i	$0.779174\pi$
$822$	−32.5369	−1.13485
$823$	−34.8013	−1.21310	−0.606549	−	0.795046i	$-0.707447\pi$
$823$	−34.8013	−1.21310	−0.606549	+	0.795046i	$0.707447\pi$
$824$	52.5854	1.83190
$825$	15.9174	0.554172
$826$	−10.6462	−0.370428
$827$	21.8038	0.758194	0.379097	−	0.925357i	$-0.376235\pi$
$827$	21.8038	0.758194	0.379097	+	0.925357i	$0.376235\pi$
$828$	1.86477	0.0648052
$829$	27.0686	0.940133	0.470066	−	0.882631i	$-0.344230\pi$
$829$	27.0686	0.940133	0.470066	+	0.882631i	$0.344230\pi$
$830$	50.5209	1.75360
$831$	−20.7653	−0.720341
$832$	−3.96786	−0.137561
$833$	−20.7192	−0.717876
$834$	32.2868	1.11800
$835$	−62.6697	−2.16877
$836$	−1.49437	−0.0516839
$837$	21.7666	0.752363
$838$	11.9153	0.411606
$839$	−29.4809	−1.01779	−0.508896	−	0.860828i	$-0.669946\pi$
$839$	−29.4809	−1.01779	−0.508896	+	0.860828i	$0.669946\pi$
$840$	27.0330	0.932726
$841$	−26.5125	−0.914225
$842$	−6.68912	−0.230522
$843$	−13.2142	−0.455123
$844$	−0.715079	−0.0246140
$845$	41.2547	1.41921
$846$	7.25914	0.249574
$847$	−12.8452	−0.441367
$848$	1.32828	0.0456135
$849$	−4.01619	−0.137835
$850$	39.1840	1.34400
$851$	−5.31373	−0.182152
$852$	8.65448	0.296497
$853$	44.8672	1.53622	0.768111	−	0.640317i	$-0.221197\pi$
$853$	44.8672	1.53622	0.768111	+	0.640317i	$0.221197\pi$
$854$	−5.20680	−0.178173
$855$	3.31246	0.113284
$856$	24.5613	0.839488
$857$	4.46945	0.152673	0.0763367	−	0.997082i	$-0.475678\pi$
$857$	4.46945	0.152673	0.0763367	+	0.997082i	$0.475678\pi$
$858$	−1.56293	−0.0533577
$859$	40.9189	1.39614	0.698068	−	0.716032i	$-0.254043\pi$
$859$	40.9189	1.39614	0.698068	+	0.716032i	$0.254043\pi$
$860$	−18.7603	−0.639723
$861$	−27.2557	−0.928871
$862$	−26.6667	−0.908273
$863$	6.91254	0.235305	0.117653	−	0.993055i	$-0.462463\pi$
$863$	6.91254	0.235305	0.117653	+	0.993055i	$0.462463\pi$
$864$	−21.6186	−0.735481
$865$	−60.9279	−2.07161
$866$	17.6699	0.600448
$867$	33.2367	1.12878
$868$	5.32900	0.180878
$869$	−16.6201	−0.563797
$870$	8.10854	0.274905
$871$	−2.23379	−0.0756891
$872$	−25.5243	−0.864361
$873$	−14.5523	−0.492522
$874$	2.88085	0.0974461
$875$	−2.63395	−0.0890438
$876$	−5.97858	−0.201998
$877$	−25.9781	−0.877217	−0.438609	−	0.898678i	$-0.644529\pi$
$877$	−25.9781	−0.877217	−0.438609	+	0.898678i	$0.644529\pi$
$878$	−37.2000	−1.25544
$879$	−23.1148	−0.779644
$880$	13.8873	0.468141
$881$	5.11415	0.172300	0.0861501	−	0.996282i	$-0.472544\pi$
$881$	5.11415	0.172300	0.0861501	+	0.996282i	$0.472544\pi$
$882$	3.77949	0.127262
$883$	10.3178	0.347221	0.173611	−	0.984814i	$-0.444457\pi$
$883$	10.3178	0.347221	0.173611	+	0.984814i	$0.444457\pi$
$884$	2.14119	0.0720160
$885$	21.9953	0.739365
$886$	9.69131	0.325586
$887$	−18.8995	−0.634582	−0.317291	−	0.948328i	$-0.602773\pi$
$887$	−18.8995	−0.634582	−0.317291	+	0.948328i	$0.602773\pi$
$888$	9.04068	0.303385
$889$	−37.3606	−1.25304
$890$	−20.5089	−0.687459
$891$	−10.1749	−0.340871
$892$	−4.53250	−0.151759
$893$	−6.24105	−0.208849
$894$	19.2791	0.644791
$895$	52.3009	1.74823
$896$	3.74516	0.125117
$897$	−1.67679	−0.0559865
$898$	9.30092	0.310376
$899$	6.06909	0.202415
$900$	3.97784	0.132595
$901$	−4.11439	−0.137070
$902$	−23.7295	−0.790105
$903$	−22.1125	−0.735858
$904$	−34.6754	−1.15329
$905$	26.9133	0.894628
$906$	−9.26619	−0.307848
$907$	10.9313	0.362967	0.181484	−	0.983394i	$-0.441910\pi$
$907$	10.9313	0.362967	0.181484	+	0.983394i	$0.441910\pi$
$908$	14.6778	0.487102
$909$	13.8051	0.457888
$910$	3.32800	0.110322
$911$	1.11377	0.0369008	0.0184504	−	0.999830i	$-0.494127\pi$
$911$	1.11377	0.0369008	0.0184504	+	0.999830i	$0.494127\pi$
$912$	−2.89211	−0.0957673
$913$	−28.8520	−0.954863
$914$	0.217681	0.00720024
$915$	10.7574	0.355629
$916$	−21.5588	−0.712324
$917$	−11.8055	−0.389851
$918$	−40.8838	−1.34937
$919$	−16.5053	−0.544459	−0.272229	−	0.962232i	$-0.587761\pi$
$919$	−16.5053	−0.544459	−0.272229	+	0.962232i	$0.587761\pi$
$920$	25.2501	0.832473
$921$	−32.6703	−1.07652
$922$	−44.6776	−1.47138
$923$	4.04538	0.133155
$924$	−4.06603	−0.133763
$925$	−11.3350	−0.372692
$926$	32.6930	1.07436
$927$	17.5321	0.575829
$928$	−6.02784	−0.197874
$929$	11.7134	0.384304	0.192152	−	0.981365i	$-0.438453\pi$
$929$	11.7134	0.384304	0.192152	+	0.981365i	$0.438453\pi$
$930$	19.7836	0.648729
$931$	−3.24942	−0.106495
$932$	−7.05523	−0.231102
$933$	−16.4057	−0.537100
$934$	−27.7560	−0.908203
$935$	−43.0162	−1.40678
$936$	−1.48301	−0.0484738
$937$	42.0470	1.37362	0.686809	−	0.726838i	$-0.259011\pi$
$937$	42.0470	1.37362	0.686809	+	0.726838i	$0.259011\pi$
$938$	10.4423	0.340952
$939$	−6.29557	−0.205448
$940$	−14.4070	−0.469903
$941$	−21.5136	−0.701322	−0.350661	−	0.936502i	$-0.614043\pi$
$941$	−21.5136	−0.701322	−0.350661	+	0.936502i	$0.614043\pi$
$942$	−12.7836	−0.416511
$943$	−25.4582	−0.829032
$944$	9.98294	0.324917
$945$	35.3637	1.15038
$946$	−19.2517	−0.625927
$947$	−57.3469	−1.86353	−0.931763	−	0.363068i	$-0.881729\pi$
$947$	−57.3469	−1.86353	−0.931763	+	0.363068i	$0.881729\pi$
$948$	7.98996	0.259502
$949$	−2.79458	−0.0907159
$950$	6.14529	0.199379
$951$	0.140062	0.00454183
$952$	−38.0046	−1.23174
$953$	47.4360	1.53660	0.768302	−	0.640087i	$-0.221102\pi$
$953$	47.4360	1.53660	0.768302	+	0.640087i	$0.221102\pi$
$954$	0.750528	0.0242992
$955$	−52.9388	−1.71306
$956$	5.43118	0.175657
$957$	−4.63072	−0.149690
$958$	38.6634	1.24916
$959$	−39.5662	−1.27766
$960$	−38.3218	−1.23683
$961$	−16.1924	−0.522334
$962$	1.11299	0.0358842
$963$	8.18879	0.263880
$964$	2.67940	0.0862977
$965$	−1.72524	−0.0555375
$966$	7.83849	0.252199
$967$	−0.219422	−0.00705613	−0.00352806	−	0.999994i	$-0.501123\pi$
$967$	−0.219422	−0.00705613	−0.00352806	+	0.999994i	$0.501123\pi$
$968$	20.4133	0.656108
$969$	8.95837	0.287784
$970$	−51.8970	−1.66631
$971$	31.0110	0.995191	0.497596	−	0.867409i	$-0.334216\pi$
$971$	31.0110	0.995191	0.497596	+	0.867409i	$0.334216\pi$
$972$	−7.24303	−0.232320
$973$	39.2620	1.25868
$974$	30.0427	0.962629
$975$	−3.57686	−0.114551
$976$	4.88243	0.156283
$977$	−20.3364	−0.650619	−0.325310	−	0.945608i	$-0.605469\pi$
$977$	−20.3364	−0.650619	−0.325310	+	0.945608i	$0.605469\pi$
$978$	28.4275	0.909012
$979$	11.7124	0.374331
$980$	−7.50103	−0.239611
$981$	−8.50985	−0.271699
$982$	−22.0932	−0.705024
$983$	−18.2190	−0.581096	−0.290548	−	0.956860i	$-0.593838\pi$
$983$	−18.2190	−0.581096	−0.290548	+	0.956860i	$0.593838\pi$
$984$	43.3140	1.38080
$985$	5.86269	0.186801
$986$	−11.3995	−0.363034
$987$	−16.9812	−0.540519
$988$	0.335806	0.0106834
$989$	−20.6542	−0.656765
$990$	7.84682	0.249388
$991$	40.9244	1.30001	0.650003	−	0.759931i	$-0.274768\pi$
$991$	40.9244	1.30001	0.650003	+	0.759931i	$0.274768\pi$
$992$	−14.7070	−0.466948
$993$	30.5997	0.971051
$994$	−18.9109	−0.599817
$995$	−17.3294	−0.549379
$996$	13.8704	0.439499
$997$	55.4741	1.75688	0.878442	−	0.477849i	$-0.158584\pi$
$997$	55.4741	1.75688	0.878442	+	0.477849i	$0.158584\pi$
$998$	40.2417	1.27383
$999$	11.8267	0.374181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.51	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.51	✓	71	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13354 q^{2} +1.40496 q^{3} -0.715079 q^{4} -3.22820 q^{5} +1.59258 q^{6} +1.93664 q^{7} -3.07766 q^{8} -1.02610 q^{9} +O(q^{10})\)	\(q+1.13354 q^{2} +1.40496 q^{3} -0.715079 q^{4} -3.22820 q^{5} +1.59258 q^{6} +1.93664 q^{7} -3.07766 q^{8} -1.02610 q^{9} -3.65931 q^{10} +2.08980 q^{11} -1.00465 q^{12} -0.469607 q^{13} +2.19527 q^{14} -4.53549 q^{15} -2.05850 q^{16} +6.37626 q^{17} -1.16313 q^{18} +1.00000 q^{19} +2.30842 q^{20} +2.72090 q^{21} +2.36888 q^{22} +2.54145 q^{23} -4.32398 q^{24} +5.42131 q^{25} -0.532321 q^{26} -5.65649 q^{27} -1.38485 q^{28} -1.57718 q^{29} -5.14117 q^{30} -3.84807 q^{31} +3.82192 q^{32} +2.93608 q^{33} +7.22777 q^{34} -6.25188 q^{35} +0.733742 q^{36} -2.09082 q^{37} +1.13354 q^{38} -0.659778 q^{39} +9.93532 q^{40} -10.0172 q^{41} +3.08425 q^{42} -8.12692 q^{43} -1.49437 q^{44} +3.31246 q^{45} +2.88085 q^{46} -6.24105 q^{47} -2.89211 q^{48} -3.24942 q^{49} +6.14529 q^{50} +8.95837 q^{51} +0.335806 q^{52} -0.645267 q^{53} -6.41188 q^{54} -6.74631 q^{55} -5.96033 q^{56} +1.40496 q^{57} -1.78780 q^{58} -4.84961 q^{59} +3.24323 q^{60} -2.37183 q^{61} -4.36196 q^{62} -1.98719 q^{63} +8.44932 q^{64} +1.51599 q^{65} +3.32817 q^{66} +4.75672 q^{67} -4.55953 q^{68} +3.57063 q^{69} -7.08677 q^{70} -8.61439 q^{71} +3.15798 q^{72} +5.95088 q^{73} -2.37004 q^{74} +7.61669 q^{75} -0.715079 q^{76} +4.04720 q^{77} -0.747887 q^{78} -7.95295 q^{79} +6.64528 q^{80} -4.86882 q^{81} -11.3549 q^{82} -13.8061 q^{83} -1.94565 q^{84} -20.5839 q^{85} -9.21222 q^{86} -2.21586 q^{87} -6.43170 q^{88} +5.60457 q^{89} +3.75482 q^{90} -0.909461 q^{91} -1.81734 q^{92} -5.40637 q^{93} -7.07450 q^{94} -3.22820 q^{95} +5.36962 q^{96} +14.1822 q^{97} -3.68336 q^{98} -2.14434 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more